Abstract
Several experts shared their views on the foundations of mathematics during International Colloquium in the Philosophy of Science held at Bedford College, Regent's Park, London, from July 11 th to 17 th 1965. They focused on the problems in the philosophy of mathematics. Various new theories constituting the foundation of mathematics are presented in the chapter. The type theory is one of the most ingeniously sophisticated systems ever produced. Intuitionism is a second important mathematical theory. Another important theory is Hilbert's proof theory––which provides a powerful method of assuring mathematicians that no new kind of logical antinomy would emerge. The proof theory is suitable for achieving negative results about the categoricity and monomorphism of formal systems. The invention of deductive inference, abstraction, and the axiomatic method was very fruitful for the development of mathematics. But it tempted mathematicians to regard mathematics as a “pure deductive science,” and to forget that their axioms were originally abstracted from empirical facts and that their rules of deductive inference are valid because, they have been tested in the actual thinking practice of mankind.
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